Introduction

The Introduction is principally addressed to the development of the canon of Heracles’ Twelve Labors. The canon is set in its Near Eastern context, with particular attention to the figure of Ninurta/Ningirsu, in the Akkadian epics Anzu and The Return of Ninurta to Nippur, and to the figure of Marduk in the Babylonian-Akkadian epic Enuma Eliš. Three chronologies are laid out: first, that of the development of the notion that Heracles had a special set of Labors as opposed to or in addition to a random series of adventures; second, that of the progression toward a more-or-less settled order for the twelve adventures eventually favored with Labor status; and, third, that of the expansion of the zone of the siting of the Labors, both within and beyond the Peloponnese. This discussion is preceded by brief material of a more general introductory nature: justification of the need for the volume and the interest of it, the articulation of its structure, and a review of recent books on Heracles.

Modeling and simulation methods of sea clutter based on measured data

The modeling and simulation of sea clutter are important in detecting radar targets in sea backgrounds. Because the nonstationary property of sea clutter is ignored in traditional statistical models, a new method based on measured sea clutter is proposed in this paper. First, we convert the measured sea clutter data under different sea conditions [[Formula: see text]] into real amplitude [Formula: see text]. Instantaneous phase [Formula: see text] is then extracted from the coherent radar’s baseband data. Second, we select a candidate statistic model and estimate its parameters based on [Formula: see text] by utilizing maximum likelihood estimation. Finally, we generate random series [Formula: see text] using corresponding random data generator and then add instantaneous phase [Formula: see text] into [Formula: see text], i.e., [Formula: see text], to obtain simulated sea clutter series. Through a comparison of simulated sea clutter and measured sea clutter data via histogram, the validity of the proposed method is proved.

Relationship Between the Structure Function of Random Time Series and the Discrete Chebyshev Spectrum

The technique of Chebyshev polynomials of discrete variable is used to analyze a random time series. The main result of the work is a new theoretical relationship between the structure function and the discrete Chebyshev spectrum. The relationship is used for the trend-resistant structural analysis of electrochemical noise of corrosion process and for the trend-resistant structural analysis of electronic noise of measuring instrument. Not only the structure function of the analyzed random series, but also the structure function of trend is estimated. The theoretical relationship between the structure function and the discrete Chebyshev spectrum can be used for the trend-resistant structural analysis of random time series of any nature.

Multiplication tables for non-prime odd numbers The untouched tables which will give us a better understanding of the distribution of prime odd numbers

The sieve method is used to separate prime numbers from non-prime numbers. If the set of prime odd numbers cannot be written as multiplication tables, the set of non-prime odd numbers can be written as multiplication tables. Thus, each odd number that does not appear in these multiplication tables is certainly a prime odd number. Based on these tables, it was proved by the opposite method that the series of prime numbers are random series. Although they are random, they can be easily tracked using the opposite method. The counter-example is used to proof that it is not possible to write whole multiplication tables of prime odd numbers on formula of [(𝑎×𝑏)+𝑐] or [(𝑎×𝑏)−𝑐]. Instead, partial multiplication tables can be used. It also was proved that the number 1 is a prime odd number.

Sufficient conditions for uniform convergence of random series

Sufficient conditions for the uniform convergence of random series are obtained.

Simulation of a Gaussian stationary process with a stable correlation function with a given reliability and accuracy

In this paper, the representation of random processes in the form of random series with uncorrelated members obtained in the work by Yu. V. Kozachenko, I.V. Rozora, E.V. Turchina (2007) [1]. Similar constructions were studied in the book by Yu. V. Kozachenko and others. [2] in the general case. However, there are additional difficulties in construction of models of specific process, such as, for example, selection of the appropriate basis in L_2(R). In this paper, models are constructed that approximate the Gaussian process with a stable correlation function $\rho_{\alpha} (h) = E X_{\alpha}(t + h) X_{\alpha}(t) = B^2 \exp{-d|h|^{\alpha}}, \alpha > 0, d > 0$ with parameter $\alpha = 2$, which is a centered stationary process with a given reliability and accuracy in the space L_p ([0,T]). And also the rates of convergence of the models are found, the corresponding theorems are formulated. Methods of representation and main properties of the process with a stable correlation function $\rho_2(h) = B^2 \exp{-d|h|^2}, d > 0$ are considered. As a basis in the space L_2(T) Hermitian functions are used.

LEBESGUE STRUCTURE OF DISTRIBUTION OF GENERALIZED BERNOULLI CONVOLUTIONS OF SPECIAL TYPE

The paper deals with the problem of deepening the Jessen-Wintner theorem for generalized Bernoulli convolutions of a special kind. The main attention is paid to the case when the terms of a random series acquire three values: 0, 1, 2. In the case when the probability that the term of a random series becomes 2 is 0, the corresponding generalized Bernoulli convolutions coincide with classic Bernoulli convolutions, which were actively studied domestic scientists (Pratsovyty M., Turbin G., Torbin G., Honcharenko Ya., Baranovsky O., Savchenko I. and others) as well as foreign researchers (Erdos P., Peres Y., Schlag W, Solomyak B., Albeverio S. and others). The problem of deepening the Jessen-Wintner theorem concerning the necessary and sufficient conditions for the distribution of a probably convergent random series with discrete additions to each of the three pure types, is extremely difficult to formulate and is not completely solved even for classical Bernoulli convolutions. The results of the study are a deepening in relation to the analysis of the Lebesgue structure of random series formed by s-expansions of real numbers. In the case when the corresponding Bernoulli convolution is generated by the sequence 3-n, we have a random variable with independent triple digits, which was studied by scientists in different directions: Lebesgue structure (Chaterji S., Marsaglia G.), topological-metric structure of the distribution spectrum (Pratsovityi M., Turbin G.), fractal analysis of the distribution carrier (Pratsovyty M., Torbin G.), asymptotic properties of the characteristic function at infinity (Honcharenko Ya., Pratsovyty M., Torbin G.). The paper presents certain sufficient conditions for the absolute continuity and singularity of the distribution, with certain restrictions on the stochastic distribution matrix and the asymptotics of the values of the random terms of the series. In the case when the Lebesgue measure of the set of realizations of the generalized Bernoulli convolution is different from zero, it is possible together with Levy's theorem to formulate criteria for belonging of the Bernoulli convolution distribution to each of the three pure Lebesgue types, namely: purely discrete, purely continuous or purely singular.